Pharmacokinetics and stochastic differential equations : model and methodology.
Maud Delattre (1), Marc Lavielle (1,2)
(1) University Paris Sud ; (2) INRIA Saclay Île-de-France
Objectives: A recent evolution of the traditional PK models based on ordinary differential equations (ODEs) consists in adding a system noise to the ODEs to account for more intra-individual variability (see [1], [2], [3]). However, the frequently proposed linear SDE system turns out to be irrelevant. First, it gives an overly erratic description of the evolution of the drug concentrations within the compartments of the human body. Second, it does not comply with some constraints on the biological dynamics (sign, monotony,...).
The objective of this contribution is to present new SDE models that would best reflect the PK reality. Some specific maximum estimation procedure for the population parameters are also developed in a population approach.
Methods: Assuming that the diffusion process randomly perturbs the transfer rate constants of the system is more realistic and allows a more accurate representation of the biological system. When it is possible to come down to a linear SDE model by some appropriate transformations of the original SDE system, we suggest estimating the population parameters by combining the SAEM algorithm with the Kalman filter. This methodology was implemented in a working version of MONOLIX and tested on some simulated basic examples.
Results: The simulated datasets show that this new model more faithfully mimics the biological dynamics. Based on these simulated examples, the proposed estimation methodology also gives encouraging results. On particular, the population parameters are estimated with little bias and the estimated standard errors for each parameter are low.
Conclusions: We have proposed a new category of mixed-effects models based on SDEs for PK modeling and our maximum likelihood estimation procedure shows quite good practical properties. We aim to extend in a next future the present approach to more complex compartment models. Defining the transfer rate constants as stochastic processes often leads to highly non linear models, in which the present estimation methodology based on the Kalman filter cannot be used. A SAEM based method using the extended Kalman filter or a particle filter should rather be considered.
References:
[1] S. Mortensen, S. Klim, B. Dammann, N. Kristensen, H. Madsen, R. Overgaard "A Matlab framework for estimation of NLME models using stochastic differential equations", Journal of Pharmacokinetics and Pharmacodynamics vol:34, pages: 623-642, 2007.
[2] R. Overgaard, E. Jonsson, C. Tornøe, H. Madsen, "Non-Linear Mixed Effects Models with Stochastic Differential Equations. Implementation of an Estimation Algorithm", PAGE 2004.
[3] Donnet S, Samson A, Parametric inference for mixed models defined by stochastic differential equations, ESAIM P&S, 12:196-218, 2008.
[4] Delattre M, Del Moral P, Lavielle M "The SAEM algorithm in MONOLIX for Non-Linear Mixed Effects Models with Stochastic Differential equations", PAGE 2010.
